An elementary matrix is one obtained by doing a single row operation to an identity matrix.
The elementary matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ results from doing the row operation ${\text{\mathbf{r}}}_{1}\leftrightarrow {\text{\mathbf{r}}}_{2}$ to ${I}_{2}$.
The elementary matrix $\left(\begin{array}{ccc}1& 2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ results from doing the row operation ${\text{\mathbf{r}}}_{1}\mapsto {\text{\mathbf{r}}}_{1}+2{\text{\mathbf{r}}}_{2}$ to ${I}_{3}$.
The elementary matrix $\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$ results from doing the row operation ${\text{\mathbf{r}}}_{1}\mapsto (-1){\text{\mathbf{r}}}_{1}$ to ${I}_{2}$.
Doing a row operation $r$ to a matrix has the same effect as multiplying that matrix on the left by the elementary matrix corresponding to $r$.
Let $r$ be a row operation and $B$ an $m\mathrm{\times}n$ matrix. Then $r\mathit{}\mathrm{(}B\mathrm{)}\mathrm{=}r\mathit{}\mathrm{(}{I}_{m}\mathrm{)}\mathit{}B$.
The theorem tells you that doing the row operation ${\mathbf{r}}_{1}\mapsto {\mathbf{r}}_{1}+2{\mathbf{r}}_{2}$ to a matrix with three rows is the same as left-multiplying by the $3\times 3$ elementary matrix $\left(\begin{array}{ccc}1& 2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$. For example, if we do this row operation to $\left(\begin{array}{c}a\\ b\\ c\end{array}\right)$ we get $\left(\begin{array}{c}a+2b\\ b\\ c\end{array}\right)$ which is the same as
$$\left(\begin{array}{ccc}1& 2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}a\\ b\\ c\end{array}\right)=\left(\begin{array}{c}a+2b\\ b\\ c\end{array}\right).$$ |
Elementary matrices are invertible.
Let $r$ be a row operation, $s$ be the inverse row operation to $r$, and let ${I}_{n}$ be an identity matrix. By Theorem 3.8.1, $r({I}_{n})s({I}_{n})=r(s({I}_{n}))$. Because $s$ is inverse to $r$, this equals ${I}_{n}$. Similarly, $s({I}_{n})r({I}_{n})=s(r({I}_{n}))={I}_{n}$. It follows that $r({I}_{n})$ is invertible with inverse $s({I}_{n})$. ∎
Theorem 3.8.1 combined with Corollary 3.8.2 shows that if $A$ results from doing a row operation to $B$, then $A=EB$ for some invertible matrix $E$. What about if $A$ results from doing a sequence of row operations?
Suppose that $A$ is a matrix obtained by doing a sequence of row operations to another matrix $B$. Then $A\mathrm{=}E\mathit{}B$ for some invertible matrix $E$.
If the row operations are ${r}_{1},\mathrm{\dots},{r}_{k}$ and if ${E}_{i}$ is the elementary matrix corresponding to ${r}_{i}$ then using Theorem 3.8.1 repeatedly gives
$$A={E}_{k}{E}_{k-1}\mathrm{\cdots}{E}_{2}{E}_{1}B.$$ |
Let $E={E}_{k}{E}_{k-1}\mathrm{\cdots}{E}_{2}{E}_{1}$, so $A=EB$. The matrix $E$ is a product of invertible matrices, by Corollary 3.8.2, so it is invertible by Theorem 3.5.3. ∎